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arXiv:2410.19541 · PRX Quantum

The Product Structure of Matrix Product States under Permutations

M. Florido-Llinas, Alvaro Martin Alhambra, Rahul Trivedi, Norbert Schuch, David Pérez-Garcı́a, J. I. Cirac

Published Nov 17, 2025• 1 citation
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Abstract

Tensor network methods have proved to be highly effective in addressing a wide variety of physical scenarios, including those lacking an intrinsic one-dimensional geometry. In such contexts, it is possible for the problem to exhibit a weak form of permutational symmetry, in the sense that entanglement behaves similarly across any arbitrary bipartition. In this paper, we show that translationally-invariant (TI) matrix product states (MPSs) with this property are trivial, meaning that they are either product states or superpositions of a few of them. The results also apply to non-TI generic MPSs, as well as further relevant examples of MPSs including the W state and the Dicke states in an approximate sense. Our findings motivate the usage of Ansätze simpler than tensor networks in systems whose structure is invariant under permutations.

DOI
10.1103/8sbs-t24w

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Tensor network methods have proved to be highly effective in addressing a wide variety of physical scenarios, including those lacking an intrinsic one-dimensional geometry. In such contexts, it is possible for the problem to exhibit a weak form of permutational symmetry, in the sense that entanglement behaves similarly across any arbitrary bipartition. In this paper, we show that translationally-invariant (TI) matrix product states (MPSs) with this property are trivial, meaning that they are either product states or superpositions of a few of them. The results also apply to non-TI generic MPSs, as well as further relevant examples of MPSs including the W state and the Dicke states in an approximate sense. Our findings motivate the usage of Ansätze simpler than tensor networks in systems whose structure is invariant under permutations.
Matrix Product States (MPS) are a particular type of one dimensional tensor network states, that have been applied to the study of numerous quantum many body problems. One of their key features is the possibility to describe and encode symmetries on the level of a single building block (tensor), and hence they provide a natural playground for the study of symmetric systems. In particular, recent works have proposed to use MPS (and higher dimensional tensor networks) for the study of systems with local symmetry that appear in the context of gauge theories. In this work we classify MPS which exhibit local invariance under arbitrary gauge groups. We study the respective tensors and their structure, revealing known constructions that follow known gauging procedures, as well as different, other types of possible gauge invariant states.
Invariant theory is concerned with functions that do not change under the action of a given group. Here we communicate an approach based on tensor networks to represent polynomial local unitary invariants of quantum states. This graphical approach provides an alternative to the polynomial equations that describe invariants, which often contain a large number of terms with coefficients raised to high powers. This approach also enables one to use known methods from tensor network theory (such as the matrix product state (MPS) factorization) when studying polynomial invariants. As our main example, we consider invariants of MPSs. We generate a family of tensor contractions resulting in a complete set of local unitary invariants that can be used to express the Rényi entropies. We find that the graphical approach to representing invariants can provide structural insight into the invariants being contracted, as well as an alternative, and sometimes much simpler, means to study polynomial invariants of quantum states. In addition, many tensor network methods, such as MPSs, contain excellent tools that can be applied in the study of invariants.
Matrix product states (MPSs) illustrate the suitability of tensor networks for the description of interacting many-body systems: ground states of gapped 1D systems are approximable by MPSs, as shown by Hastings [M. B. Hastings, J. Stat. Mech. (2007) P08024]. By contrast, whether MPSs and more general tensor networks can accurately reproduce correlations in critical quantum systems or quantum field theories has not been established rigorously. Ample evidence exists: entropic considerations provide restrictions on the form of suitable ansatz states, and numerical studies show that certain tensor networks can indeed approximate the associated correlation functions. Here, we provide a complete positive answer to this question in the case of MPSs and 2D conformal field theory: we give quantitative estimates for the approximation error when approximating correlation functions by MPSs. Our work is constructive and yields an explicit MPS, thus providing both suitable initial values and a rigorous justification of variational methods.
Tensor networks (TNs) are one of the best available tools to study many-body quantum systems. TNs are particularly suitable for one-dimensional local Hamiltonians, while their performance for generic geometries is mainly limited by two aspects: the limitation in expressive power and the approximate extraction of information. Here we investigate the performance of superposition-of-product-states (SPS) ansatz, a variational framework structurally related to canonical polyadic tensor decomposition. The ansatz does not compress information as effectively as tensor networks, but it has the advantages (i) of allowing accurate extraction of information, (ii) of being structurally independent of the geometry of the system, (iii) of being readily parallelizable, and (iv) of allowing analytical shortcuts. We first study the typical properties of the SPS ansatz for spin-1/21/21/2 systems, including its entanglement entropy, and its trainability. We then use this ansatz for ground state search in tilted Ising models -- including one-dimensional and three-dimensional with short- and long-range interaction, and a random network -- demonstrating that SPS can attain high accuracy.
We determine the local symmetries and local transformation properties of certain many-body states called translationally invariant matrix product states (TIMPSs). We focus on physical dimension d=2d=2d=2 of the local Hilbert spaces and bond dimension D=3D=3D=3 and use the procedure introduced in Sauerwein et al. [Phys. Rev. Lett. 123, 170504 (2019)] to determine all (including nonglobal) symmetries of those states. We identify and classify the stochastic local operations assisted by classical communication (SLOCC) that are allowed among TIMPSs. We scrutinize two very distinct sets of TIMPSs and show the big diversity (also compared to the case D=2D=2D=2) occurring in both their symmetries and the possible SLOCC transformations. These results reflect the variety of local properties of MPSs, even if restricted to translationally invariant states with low bond dimension. Finally, we show that states with nontrivial local symmetries are of measure zero for d=2d=2d=2 and D>3D>3D>3.
We analyze entanglement in the family of translationally invariant matrix product states (MPS). We give a criterion to determine when two states can be transformed into each other by local operations with a nonvanishing probability, a central question in entanglement theory. This induces a classification within this family of states, which we explicitly carry out for the simplest, nontrivial MPS. We also characterize all symmetries of translationally invariant MPS, both global and local (inhomogeneous). We illustrate our results with examples of states that are relevant in different physical contexts.